Galerkin‐based mixed time finite elements for structural dynamics

Abstract

AbstractNumerically stable and accurate time integration methods are important in structural dynamics. We propose such a method based on a mixed variational formulation of the underlying semi‐discrete equations of motion. The newly developed scheme relies on a specific combination of the so‐called continuous Galerkin (cG(k)) method and the discontinuous Galerkin (dG(k)) method. We refer to [2] for an introduction of the original cG(k) and dG(k) methods for ODEs (see also [1]). The mixed approach relies on polynomial approximations of degree k on the level of each time finite element. In particular we make use of hierarchical interpolation polynomials. The present approach is closely related to Hellinger‐Reissner type mixed variational formulations for elastodynamics. The mixed method yields implicit time‐stepping schemes with order of accuracy equal to 2k for the displacements and velocities. In addition to that, the method is capable of conserving the energy in the case of conservative mechanical systems. Numerical examples are presented which highlight the numerical properties of the newly proposed mixed approach.